ENGG2013 Unit 10n n determinant and

an application to cryptographyFeb, 2011.

Yesterday – A formula for matrix inverse using cofactors

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Suppose that det A is nonzero.

Three steps in computing above formula1. for i,j = 1,2,3, replace each aij by cofactor Cij

2. Take the transpose of the resulting matrix.3. divide by the determinant of A.

Usually called the adjoint of A

cofactors

Outline

• nxn determinant• Caesar Cipher• Modulo arithmetic• Hill Cipher

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DETERMINANT IN GENERAL

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A pattern

• Arrange the products so that the first subscripts are in ascending order.• All possible orderings of the second subscripts appear once and only once.

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Transposition

• A transposition is an exchange of two objects in a list of objects.

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A B C D

A C B D

Examples:

2 1 4 5 3

1 2 4 5 3

“Transposition” is anothermathematical term, and isnot the same as matrix tranpose.

Another pattern

• The sign of each term is closely related to the number of transpositions required to obtain the second subscripts, starting from (1,2) for the 2x2 case or (1,2,3) for the 3x3 case.

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The sign

• Let p(1), p(2), …, p(n) be an order of 1,2,…,n.– For example p(1)=3, p(2) = 2, p(3)=1 is an ordering

of 1, 2, 3.

• Starting from (1,2,…,n), if we need an odd no. of transpositions to get ( p(1), p(2), …, p(n) ), we define the sign of (p(1), p(2),…,p(n)) be –1.

• Otherwise, if we need an even no. of transpositions to get ( p(1), p(2), …, p(n) ), we define the sign of (p(1), p(2),…,p(n)) be +1.

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Definition of nn determinant

• The summation is over all n! possible orderings p = ( p(1), p(2), …, p(n) ) of 1,2,…,n.– There are n! terms.

• sgn(p) is either +1 or –1, usually called the signature or signum of p.

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http://en.wikipedia.org/wiki/Determinant

1

Properties of determinant

• Determinant of nn identity matrix equals 1.• Exchange two rows (or columns) multiply

determinant by –1.• Multiply a row (or a column) by a constant k

multiply the determinant by k.• Add a constant multiple of a row (column) to

another row (column) no change• Additive property as in the 33 and 22 case.

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Cofactor and the adjoint formula for matrix inverse

• Cofactors are defined in a similar way as in the 3x3 case.– The cofactor of the (i,j)-entry of a matrix A, denoted by Cij, is

defined as (–1)i+j Aij, where A is the determinant of the sub-matrix obtained by removing the i-th row and the j-th column.

• We have similar expansion along a row or a column (also called the Laplace expansion) as in the 3x3 case.

• The adjoint formula:

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nxn identityA adjoint of A

The formula in this form holds when det A = 0 also

transpose

CAESAR CIPHER

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Caesar and his army

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ATTACK

Soldier carrying themessage “ATTACK”

Message may be interceptedby enemy

Caesar cipher

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http://en.wikipedia.org/wiki/Caesar_cipher

ATTACK

Soldier carrying theencrypted message“DWWDFN”

The encrypted messagelooks random and meaningless

Private key encryption

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Plain text Encryptionfunction Ciphertext

Plain text Decryptionfunction Ciphertext

Key

key

The value of “key” is keptsecret

Mathematical description

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ATTACK Shift to the rightby 3 DWWDFN

ATTACKShift to the left

by 3 DWWDFN

Key =3

Key = 3

Caesar cipher is not secureenough, because the numberof keys is too small.

MODULO ARITHMETIC

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Mod 12

• Clock arithmetic

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121

2

9 3

6

4

57

8

10

11

6+8= 2 mod 12

5+12 = 5 mod 12

Mod 7

• Week arithmetic

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6

1+9 = 3 mod 7

2+3 = 5 mod 7

Sun Mon Tue Wed Thr Fri Sat

1

2 3 4 5 6 7 8

9 10 11 12 13 14 15

16 17 18 19 20 21 22

23 24 25 26 27 28 29

30 31

0 1 2 3 4 5 6

Mod 60• 天干地支 arithmetic

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Year of rabbit

Mod n – formal definition

• n is a fixed positive integer• Definition: a mod n is the remainder of a after

division by n.– Example: 25 = 1 mod 12.

• Addition and multiplication: If the sum or product of two integers is larger than or equal to n, divide by n and take the remainder.– Example: 2+10 = 0 mod 12.– Example: 25 = 3 mod 12.

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More examples

• 10 mod 7 = 3• 4+5 mod 7 = 2• 6+7 mod 7 = 6• 27 mod 7 = 0

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Mod 26

A B C D E F G H I J K L M

0 1 2 3 4 5 6 7 8 9 10 11 12

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N O P Q R S T U V W X Y Z

13 14 15 16 17 18 19 20 21 22 23 24 25

Fix a one-to-one correspondence between the English alphabetsand the integers mod 26.

Caesar’s cipher: shifting a letter to the right by 3is the same as adding 3 in mod 26 arithmetic.

Examples of mod 26 calculations

• 3+19 = ? mod 26• 13+20 = ? mod 26• 34 = ? Mod 26• 134 = ? Mod 26

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A B C D E F G H I J K L M

0 1 2 3 4 5 6 7 8 9 10 11 12

N O P Q R S T U V W X Y Z

13 14 15 16 17 18 19 20 21 22 23 24 25

Peculiar phenomena in modulo arithmetic

• Non-zero times non-zero may be zero– 49 = 0 mod 12– 22 = 0 mod 4

• Multiplicative inverse may not exist– Cannot find an integer x such that 4x = 1 mod 12.

4-1 does not exist mod 12.

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No fraction in modulo arithmetic• In mod 12, don’t write 1/3 or 3-1 because it does not exist. • But 5-1 is well-defined mod 12, because we can solve 5x=1 mod 12.

Indeed, we have 55 = 1 mod 12. Therefore 5-1 = 5 mod 12.

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FractionFact from number theory: multiplicative inverse of x mod n existsif and only the gcd of x and n is 1.

HILL CIPHER

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Hill cipher• Invented by L. S. Hill in 1929.• Inputs : String of English letters, A,B,…,Z.

An nn matrix K, with entries drawn from 0,1,…,25.(The matrix K serves as the secret key. )

• Divide the input string into blocks of size n.• Identify A=0, B=1, C=2, …, Z=25.• Encryption: Multiply each block by K and then

reduce mod 26.• Decryption: multiply each block by the inverse of

K, and reduce mod 26.

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http://en.wikipedia.org/wiki/Hill_cipher

Note

• The decryption must be the inverse function of the encryption function.– It is required that K-1 K = In mod 26.

• Provided that det(K) has a multiplicative inverse mod 26, i.e., if det(K) and n has no common factor, the inverse of K can be computed by the adjoint formula for matrix inverse.

• Inverse of an integer mod 26 can be obtained by trial and error.

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Example• Plain text: “LOVE”, Secret Key:• “LO”

• “VE” • 2, 3, 16, 5 are transformed to cipher text “CDQF”

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A B C D E F G H I J K L M

0 1 2 3 4 5 6 7 8 9 10 11 12

N O P Q R S T U V W X Y Z

13 14 15 16 17 18 19 20 21 22 23 24 25

How to decode?

• Given “CDQF”, and the encryption matrix• How do we decrypt?

– We need to compute the inverse of

• Remind that all arithmetic are mod 26. There is no fraction and care should be taken in computing multiplicative inverse mod 26.

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Determinant

• The determinant of equals 20(7)-3(15),which is 17 mod 26.

• Find the multiplicative inverse of 17 mod 26, i.e., find integer x such that 17x = 1 mod 26.

• Just try all 26 possibilities for x:

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171 = 17 mod 26172= 8 mod 26173 = 25 mod 26174 = 16 mod 26175 = 7 mod 26176 = 24 mod 26177 = 15 mod 26

178 = 6 mod 26179= 23 mod 261710 = 14 mod 261711 = 5 mod 261712 = 22 mod 261713 = 13 mod 261714 = 4 mod 26

1715 = 21 mod 261716= 12 mod 261717 = 3 mod 261718 = 20 mod 261719 = 11 mod 261720 = 2 mod 261721 = 19 mod 26

1722 = 10 mod 261723= 1 mod 261724 = 18 mod 261725 = 9 mod 26170 = 0 mod 26

Computing the inverse mod 26

• From 1723= 1 mod 26, we know that the multiplicative inverse of 17 mod 26 is 23.

• Using the formula for 2 2 matrix inverse

we get

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Replace (17)-1 mod 26 by 23

Decryption

• Given the ciphertext “CDQF”, we decrypt by multiplying by

• From the table in p.23, 11, 14, 21, 4 is “LOVE”. kshum ENGG2013 34